Automorphisms of certain design groups. II (Q2370272)

A design group \((N,\mathbf{B}_\phi,+)\) is a \(2\)-design \((N,\mathbf{B}_\phi)\) arising by the Ferrero pair \((N,\phi)\) where the right translations of \(N\) act as automorphisms of the design \((N,\mathbf{B}_\phi)\). Let \(f:(M, \mathbf{B}_\psi,+)\rightarrow (N,\mathbf{B}_\phi,+)\) be an isomorphism. For \(k=| \phi | \) if \(| N/[N,N]| >2k^2-6k+1\) then \(f\psi f^{-1}=\phi\). In particular \(\text{Aut} (N,\mathbf{B}_\phi)\) is the normalizer of \(\phi\) inside \(\text{Aut} (N,+)\). The bound is also discussed, e.g. showing an interesting class of Ferrero pairs nonfulfilling the conclusion of the theorem but giving identical designs. It is pleasant to read the complex proofs. [For Part I see J. Algebra 167, No.~2, 488--500 (1994; Zbl 0806.05023).]